Method and Device for Synchronizing Rectilinear or Quasi-Rectilinear Links in the Presence of Interference

ABSTRACT

A method of synchronizing a substantially rectilinear signal being propagated through an unknown channel, in the presence of unknown substantially rectilinear interferences, received by an array of N sensors, in which a known training sequence s(nT) is used comprising K symbols and sampled at the symbol rate T (s(nT), 0≦n≦{tilde over (K)}1), characterized in that, based on observations x((n+l/p)T) over the duration of the training sequence, where p=T/Te is an integer and Te the sampling period, a virtual observation vector X((n+l/p)T)=[x((n+l/p)T) T , x((n+l/p)T) † ] T  is defined, as well as a decision criterion or decision statistic taking into account the second-order non-circular nature of the interferences, by using the first and second correlation matrices of the virtual observation vector X((n+l/p)T).

CROSS REFERENCE TO RELATED APPLICATIONS

The present Application is based on International Application No.PCT/EP2006/060102, filed Feb. 20, 2006 which in turn corresponds toFrance Application No. 05 01784, filed Feb. 22, 2005, and priority ishereby claimed under 35 USC §119 based on these applications. Each ofthese applications are hereby incorporated by reference in theirentirety into the present application.

FIELD OF THE INVENTION

The invention relates notably to a method and a device for synchronizinga rectilinear or quasi-rectilinear link in the presence of interferencesof the same type, from one or more receiving antennas.

It can be used notably to synchronize, from the same antenna, a linkdisturbed by interference, the interference possibly being of the sametype.

In this description, the term “rectilinear link” is used to denote alink for which the transmitted signal is rectilinear, that is, that ithas a real complex envelope, or a one-dimensional modulation, as is thecase for signals with amplitude modulation AM, Amplitude Shift KeyingASK or binary phase shift keying BPSK modulation. A link is said to bequasi-rectilinear if the transmitted signal is quasi-rectilinear, thatis, if the real part of its complex envelope contains all theinformation conveyed by this signal. These quasi-rectilinear signalsinclude, in particular, the MSK or GMSK signals having been subjected toa derotation preprocessing operation.

BACKGROUND OF THE INVENTION

The invention applies notably to systems using modulations that arerectilinear, or rendered quasi-rectilinear after preprocessing, such ascertain friend-foe identification systems (IFF in modes S and 5) or evencertain radio communication networks (cellular or otherwise) such as theGlobal System for Mobile communications GSM, for which the main sourceof interferences is the network itself.

The problem of synchronizing the links in the presence of interferenceis a problem that has been given enormous attention over the last twodecades, mainly in the fight against co-channel interference in thecontext of multiple-access networks relying on a code or CDMA(code-division multiple access).

These techniques operate on the basis of one [1-2] or several [3-5]antennas [9] in reception. However, the single-sensor techniques are allvery specific to the CDMA context and cannot be considered in F/TDMAnetworks (networks with multiple or time-division multiple access, withslow (or quick) time-frequency hopping (F-TDMA)). Also, themultiple-sensor techniques proposed in [4] [9] are similar whereas thosederived from [3] also remain specific to CDMA networks in as much as thespreading codes are assumed to be not modulated randomly by informationsymbols. In fact, only the maximum likelihood approach proposed in [5]can be envisaged outside the CDMA context. However, this approachpresupposes stationary Gaussian interferences, which are thereforesecond-order circular, and does not use any a priori particular ones ofthe interferences. In particular, the approach proposed in [5] becomessub-optimal in the presence of second-order non-circular interferences,for which the second correlation function is not identically zero, aproperty characteristic of the GMSK signals used by the GSM networks inparticular, which become quasi-rectilinear after preprocessing.

SUMMARY OF THE INVENTION

The subject of the invention relates notably to a method and a devicefor synchronizing a link that is rectilinear or renderedquasi-rectilinear after preprocessing, exploiting the potentiallynon-circular nature of the interferences and particularly powerful forso-called internal interferences which are themselves rectilinear orquasi-rectilinear after preprocessing.

The idea of the invention is notably to exploit the knowledge of atraining sequence included in the bursts of the link for synchronizationpurposes in particular, and applies an optimal widely-linear filteringto the observations. Remember that a widely-linear filtering is acombined linear filtering of the observations and of the conjugatecomplex observations [7].

The invention relates to a method of synchronizing a substantiallyrectilinear signal being propagated through an unknown channel, in thepresence of unknown substantially rectilinear interferences, received byan array of N sensors, in which a known training sequence s(nT) is usedcomprising K symbols and sampled at a symbol rate T (s(nT), 0≦n≦K−1)characterized in that, based on observations x((n+l/p)T) over theduration of the training sequence, where p=T/Te is an integer and Te thesampling period, a virtual observation vectorX((n+l/p)T)=[x((n+l/p)T)^(T), x((n+l/p)T)^(†)]^(T) is defined, as wellas a decision criterion or decision statistic taking into account thesecond-order non-circular nature of the interferences, by using thefirst and second correlation matrices of the virtual observation vectorX((n+l/p)T).

For rectilinear signals, the method comprises, for example, thefollowing steps:

-   -   acquiring observation vectors (N×1), x((l/p+n)T), 0≦n≦K−1, where        T=pT_(e) is the symbol duration, p is an integer and T_(e) the        sampling period, 1 being the delay of the signal,    -   constructing virtual observation vectors (2N×1),        X((n+l/p)T)=[x((n+l/p)T)^(T), x((n+l/p)T)^(†)]^(T),    -   choosing L as the number of time coefficients and constructing        virtual space-time observation vectors (2LN×1),        X_(st)((n+l/p)T)Δ[X((l/p+n+(L−1)/2)T)^(T), . . . ,        X((l/p+n−(L−1)/2)T)^(T)]^(T) if L is odd and        X_(st)((l/p+n)T)Δ[X((l/p+n+L/2)T)^(T), . . . ,        X((l/p+n−L/2+1)T)^(T)]^(T) if L is even,    -   determining the intercorrelation vector·{circumflex over        (r)}_(X,st,s)(l) and the correlation matrix {circumflex over        (R)}_(X,st)(l) of the space-time observations, for the sampling        instant l, taking into account the second correlation matrix of        the vector x((l/p+n)T),    -   defining a space-time filter ST and its estimate from        {circumflex over (R)}_(X,st)(l) and {circumflex over        (r)}_(X,st,s)(l), such that Ŵ_(st)(l)Δ{circumflex over        (R)}_(X,st)(l)⁻¹ {circumflex over (r)}_(X,st,s)(l),    -   defining a synchronization criterion Ĉ_(NCIR-LR)(l) by        correlating the output of the space-time filter Ŵ_(st)(l) and        the training sequence,    -   comparing the criterion Ĉ_(NCIR-LR)(l) with a threshold β set        for a given false alarm probability.

The method, for two-state CPM signals, comprises a preprocessing stepfor derotating the received observations in order in particular torender the signal quasi-rectilinear.

The decision criterion is, for example, obtained as follows:

${{\hat{C}}_{{NCIR}\text{-}{LR}}(l)}\underset{\_}{\Delta}\frac{{{\hat{r}}_{X,{st},s}(l)}^{\dagger}{{\hat{R}}_{X,{st}}(l)}^{- 1}{{\hat{r}}_{X,{st},s}(l)}}{\left( {1/K} \right){\sum\limits_{n = 0}^{K - 1}{{s({nT})}}^{2}}}$

where the matrix (2LN×2LN) {circumflex over (R)}_(X,st)(l) and thevector (2LN×1) {circumflex over (r)}_(X,st,s)(l) are definedrespectively by:

${{\hat{R}}_{X,{st}}(l)}\underset{\_}{\Delta}\frac{1}{K}{\sum\limits_{n = 0}^{K - 1}{{X_{st}\left( {\left( {{l/p} + n} \right)T} \right)}{X_{st}\left( {\left( {{l/p} + n} \right)T} \right)}^{\dagger}}}$${{\hat{r}}_{X,{st},s}(l)}\underset{\_}{\Delta}\frac{1}{K}{\sum\limits_{n = 0}^{K - 1}{{X_{st}\left( {\left( {{l/p} + n} \right)T} \right)}{s({nT})}^{*}}}$

where 0≦Ĉ_(NCIR-LR)(l)≦1.

The invention also relates to a device for synchronizing a substantiallyrectilinear signal being propagated through an unknown channel, in thepresence of unknown substantially rectilinear interferences, in an arrayof N sensors, in which a known training sequence s(nT) is usedcomprising K symbols and sampled at the symbol rate (s(nT), 0≦n≦K−1),characterized in that it comprises a device able to determine a decisioncriterion or decision statistic from observations x((n+l/p)T) over theduration of the training sequence, where p=T/Te is an integer and Te thesampling period, a virtual observation vectorX((n+l/p)T)=[x((n+l/p)T)^(T), x((n+l/p)T)^(†)]^(T), and a decisioncriterion or decision statistic, taking into account the second-ordernon-circular nature of the interferences, by using the first and secondcorrelation matrices of the virtual observation vector X((n+l/p)T).

The device comprises, for example, an array of virtual sensors (N+1 to2N), a filter (1), a device for correlating the signal obtained from thefilter and from the training sequence, and a decision device receivingthe correlated signal.

It can also comprise a device able to convert a two-state CPM signalinto a quasi-rectilinear signal.

The method and the device are used, for example, to synchronize aone-dimensional modulation signal: ASK, BPSK, etc. or a two-state CPMsignal of MSK, GMSK or other type.

The invention has the particular advantages of reducing, with constantperformance levels, the number of sensors in reception and also makes itpossible to envisage synchronization from a single antenna in thepresence of an interference.

BRIEF DESCRIPTION OF THE DRAWINGS

Other characteristics and advantages of the present invention willbecome more apparent from reading the description that follows of anillustrative and by no means limiting example with appended figures,which represent:

FIG. 1, a functional diagram of the synchronization device according tothe invention,

FIG. 2, configuration examples for optimal synchronization in thepresence of a BPSK interference,

FIG. 3, examples of useful and interfering constellations at the outputof the filter.

FIG. 1 represents a block diagram of an exemplary device according tothe invention in the case of a BPSK modulation comprising: an array of Nsensors referenced 1 to N in the figure, N receiving channels, a virtualarray of sensors referenced N+1 to 2N in the figure, a filter 1, a means2 for correlating a reference signal with the signal obtained from thefilter, and a decision device 3 (detection/synchronization).

The filter 1 is, for example, a LSL Wiener filter, with the particularfunction of rejecting interferences and rejecting decorrelated paths.

The decision device 3 is for calculating the sufficient statistic andcomparing this statistic with a fixed threshold. These mechanisms aredescribed below.

This example is given as an illustration in order to understand theinvention. It is obvious that this scheme can be modified and adapted tothe processing of AM, ASK, MSK or GMSK, and other signals. This list isgiven as an illustration.

Before detailing how the steps of the method according to the inventionare implemented, some recaps and assumptions to enable it to beunderstood are explained.

An antenna with N narrowband (NB) sensors is considered, the antennareceiving the contribution of a rectilinear useful source, assumed to beBPSK (binary phase shift keying) modulated for simplification purposes,and a total noise consisting of rectilinear interferences and backgroundnoise. Given these assumptions, the vector, x(kT_(e)), of the complexenvelopes of the signals sampled and observed at the output of thesensors is given by:

x(kT _(e))≈s((k−l _(o))T _(e))h _(s) +b _(T)(kT _(e))  (1)

where T_(e) is the sampling period, b_(T)(kT_(e)) to the sampled totalnoise vector, uncorrelated with the useful source, h_(s) is the vectorof the impulse responses of the channels associated with the usefulsignal, l_(o) is the propagation delay of the useful signal, assumed tobe equal to a multiple of T_(e) in the interests of simplicity, and s(t)is the complex envelope of the BPSK useful signal given by:

$\begin{matrix}{{s(t)} = {\mu_{s}{\sum\limits_{n}{a_{n}{v\left( {t - {nT}} \right)}}}}} & (2)\end{matrix}$

where a_(n)=±1 are random variables that are independent and identicallydistributed (i.i.d) corresponding to the transmitted signals, T is thesymbol duration, assumed to be such that T=pT_(e), where p is aninteger, v(t) is the raised cosinusoidal formatting filter (½ Nyquistfilter) and μ_(s) is a real value controlling the instantaneous power ofs(t). It should be noted that the model (1) assumes propagation channelswithout time spreading, which is produced, for example, for apropagation in free space (satellite telecommunications, airportradiocommunications) or channels with flat fading (certainradiocommunication situations in an urban area). In particular, for apropagation in free space, h_(s) Δe^(jφs)s where φ_(s) and srespectively correspond to the phase and the controlling vector of theuseful signal.

However, the method explained in this document also applies totime-spread channels. In these conditions, the useful part of (1) infact corresponds to the contribution of a path or propagation mode. Theother paths are in the total noise vector.

Second-Order Statistics of the Observations

The second-order statistics, considered hereinafter to correspond to thefirst, R_(X)(k), and the second, C_(X)(k), correlation matrices ofx(kT_(e)), defined, given the above assumptions, by:

R _(X)(k)Δ E[x(kT _(e))x(kT _(e))^(†)]≈π_(s)(k−l _(o))h _(s) h _(s) ^(†)+R(k)  (3)

C _(X)(k)Δ E[x(kT _(e))x(kT _(e))^(T)]≈π_(s)(k−l _(o))h _(s) h _(s) ^(T)+C(k)  (4)

which depend on the time in as much as the BPSK useful signal is acyclostationary signal and the total noise is also assumed to becyclostationary with the same cyclical frequencies as the useful signal,which is in particular the case in the presence of internalinterferences. In the above expressions, the sign ^(†) signifiesconjugate transpose, R(k)ΔE[b_(T)(kT_(e)) b_(T)(kT_(e))^(†)] andC(k)ΔE[b_(T)(kT_(e))b_(T)(kT_(e))^(T)] are respectively the first andsecond correlation matrices of the noise vector, b_(T)(kT_(e)),π_(s)(k)ΔE[|s(kT_(e))|²] is the instantaneous power of the useful signalreceived by an omnidirectional sensor for a propagation in free space.

Statement of the Problem

In a radiocommunication system, training sequences are normallytransmitted periodically for synchronization purposes, which means inparticular that the useful signal s(kT_(e)) is known over intervals witha duration of K symbols, where pK is the number of samples of thetraining sequence. In such a context, assuming R(k), C(k) and h_(s) tobe unknown, the problem of optimal synchronization is to find the bestestimate, ĺ_(o), of l_(o) from the observation vectors x(kT_(e)) andknowing the useful signal s(kT_(e)) for 0≦k≦pK−1. This problem is alsoequivalent to finding l=ĺ_(o) such that the known samples s(kT_(e)),0≦k≦pK−1, are optimally detected from the observation vectorsx((k+l)T_(e)), 0≦k≦pK−1.

By taking the optimal synchronization instant l_(o)T_(e) and consideringthe situation with two assumptions:

H0: presence of only total noise in x((k+l_(o))T_(e)), and H1: presenceof total noise and the useful signal in x((k+l_(o))T_(e)), the followingcan be stated:

H1: x((k+l _(o))T _(e))≈s((kT _(e))h _(s) +b _(T)((k+l _(o))T_(e))  (5a)

H0: x((k+l _(o))T _(e))≈b _(T)((k+l _(o))T _(e))  (5b)

In this context, according to the statistical theory of detection (inthe Neyman Pearson sense) [8], the optimal strategy for detecting theuseful signal s(kT_(e)) from observations x((k+l_(o))T_(e)) over theduration of the training sequence, is to compare with a threshold, thelikelihood ratio (RV), L(x)(l₀), defined by:

$\begin{matrix}{{L(x)}\left( l_{o} \right)\underset{\_}{\Delta}\frac{p\left\lbrack {{x\left( {\left( {k + l_{o}} \right)T_{e}} \right)},{0 \leq k \leq {{pK} - 1}},{{/H}\; 1}} \right\rbrack}{p\left\lbrack {{x\left( {\left( {k + l_{o}} \right)T_{e}} \right)},{0 \leq k \leq {{pK} - 1}},{{/H}\; 0}} \right\rbrack}} & (6)\end{matrix}$

where p[x((k+l_(o))T_(e)), 0≦k≦pK−1/Hi] (i=0, 1) is the conditionalprobability density of the vector [x(l_(o)T_(e))^(T),x((1+l_(o))T_(e))^(T), . . . , x((pK+l_(o)−1)T_(e))^(T)]^(T) given theassumption Hi.

The idea of the invention is notably to exploit the second-ordernon-circular nature of the interferences while keeping the assumption oftotal Gaussian noise and that of the stationarity of the total noise,retaining only one total noise sample for each symbol over the durationof the sequence.

This means that, over the duration of the training sequence, only thenoise vectors b_(T)((l_(o)/p+n)T), 0≦n≦K−1 are considered.

Given these assumptions, the probability density of the total noisevector b_(T)((l_(o)/p+n)T) is given by:

p[B_(T)((l_(o)/p+n)T)]Δπ^(−N)det[R_(B)(l_(o))]^(−1/2)exp[−(½)B_(T)((l_(o)/p+n)T)^(\)R_(B)(l_(o))⁻¹B_(T)((l_(o)/p+n)T)]  (7)

where B_(T)((l_(o)/p+n)T) is the vector (2N×1) defined byB_(T)((l_(o)/p+n)T) Δ[b_(T)((l_(o)/p+n)T)_(T),b_(T)((l_(o)/p+n)T)^(†)]^(T), and where R_(B)(l_(o)) is the matrix(2N×2N) defined by:

$\begin{matrix}{{{R_{B}\left( l_{o} \right)}\underset{\_}{\Delta}{E\left\lbrack {{B_{T}\left( {\left( {{l_{o}/p} + n} \right)T} \right)}\mspace{14mu} {B_{T}\left( {\left( {{l_{o}/p} + n} \right)T} \right)}^{\dagger}} \right\rbrack}} = \begin{pmatrix}{R\left( l_{o} \right)} & {C\left( l_{o} \right)} \\{C\left( l_{o} \right)}^{*} & {R\left( l_{o} \right)}^{*}\end{pmatrix}} & (8)\end{matrix}$

In these conditions, assuming the vectors B_(T)((l_(o)/p+n)T) for0≦n≦K−1, to be uncorrelated, the likelihood ratio RV, L(x)(l_(o)),defined by (6), becomes:

$\begin{matrix}{{L(x)}\left( l_{o} \right)\underset{\_}{\Delta}\frac{\begin{matrix}{\prod\limits_{n = 0}^{K - 1}{p\left\lbrack {{B_{T}\left( {\left( {{l_{o}/p} + n} \right)T} \right)} =} \right.}} \\\left. {{{X\left( {\left( {{l_{o}/p} + n} \right)T} \right)} - {{s({nT})}{H_{s}/{s({nT})}}}},H_{s},{R_{B}\left( l_{o} \right)}} \right\rbrack\end{matrix}}{\prod\limits_{n = 0}^{K - 1}{p\left\lbrack {{B_{T}\left( {\left( {{l_{o}/p} + n} \right)T} \right)} = {{X\left( {\left( {{l_{o}/p} + n} \right)T} \right)}/{R_{B}\left( l_{o} \right)}}} \right\rbrack}}} & (9)\end{matrix}$

where the vectors (2N×1) X((l_(o)/p+n)T) and H_(s) (extended propagationchannel vector) are respectively defined by X((l_(o)/p+n)T)Δ[x((l_(o)/p+n)T)^(T), x((l_(o)/p+n)T)^(†)]^(T) and H_(s) Δ[h_(s) ^(T),h_(s) ^(\)]^(T). The quantities H_(s) and R_(B)(l_(o)) are assumed to beunknown and must be replaced in (9) by their estimate in the maximumlikelihood sense. In these conditions, it can be shown that, after a fewmathematical manipulations of the expression (9), a sufficient statisticfor the optimal detection of the sequence s(nT) from the vectorsx((l_(o)/p+n)T), over the duration of the sequence, 0≦n≦K−1, is, for asecond-order non-circular total noise, given by:

$\begin{matrix}{{{\hat{C}}_{{NCIR}\text{-}{LR}}\left( l_{o} \right)}\underset{\_}{\Delta}\frac{{{\hat{r}}_{Xs}\left( l_{o} \right)}^{\dagger}{{\hat{R}}_{X}\left( l_{o} \right)}^{- 1}{{\hat{r}}_{Xs}\left( l_{o} \right)}}{\left( {1/K} \right){\sum\limits_{n = 0}^{K - 1}{{s({nT})}}^{2}}}} & (10)\end{matrix}$

where the vector {circumflex over (r)}_(Xs)(l_(o)) and the matrix{circumflex over (R)}_(X)(l_(o)) are given by:

$\begin{matrix}{{{\hat{r}}_{Xs}\left( l_{o} \right)}\underset{\_}{\Delta}\frac{1}{K}{\sum\limits_{n = 0}^{K - 1}{{X\left( {\left( {{l_{o}/p} + n} \right)T} \right)}{s({nT})}^{*}}}} & (11) \\{{{\hat{R}}_{X}\left( l_{o} \right)}\underset{\_}{\Delta}\frac{1}{K}{\sum\limits_{n = 0}^{K - 1}{{X\left( {\left( {{l_{o}/p} + n} \right)T} \right)}{X\left( {\left( {{l_{o}/p} + n} \right)T} \right)}^{\dagger}}}} & (12)\end{matrix}$

where 0≦Ĉ_(NCIR-LR)(l_(o))≦1.

It can be deduced from the above results that the optimalsynchronization strategy with non-circular Gaussian total noise, calledoptimal strategy in this document, consists in calculating, at eachsampling instant lT_(e), the expression Ĉ_(NCIR-LR)(l), defined by (10)where l replaces l_(o), and comparing the result with a threshold, whichis set for a given false alarm probability. The optimal synchronizationinstant then corresponds to the instant lT_(e)=ĺ_(o)T_(e) such thatĺ_(o) generates the maximum value of Ć_(NCIR-LR)(l) out of those thatexceed the threshold.

The synchronization method for the rectilinear signals comprises, forexample, the following steps:

-   -   Step 0: Initialization l=l_(min) (l_(min)=0 for example) and        choice of the detection threshold β    -   Step 1: Estimation of {circumflex over (r)}_(Xs)(l) and of        {circumflex over (R)}_(X)(l)    -   Step 2: Calculation of the sufficient statistic Ĉ_(NCIR-LR)(l)    -   Step 3: Comparison of Ĉ_(NCIR-LR)(l) with the threshold β    -   Step 4: Decision        -   If Ĉ_(NCIR-LR)(l)<β            -   If Ĉ_(NCIR-LR)(l−1)<β                -   l=l+1                -   return to step 1            -   If Ĉ_(NCIR-LR)(l−1)≧β                -   The synchronization instant is ĺ_(o)T_(e) where                    l=ĺ_(o) maximizes Ĉ_(NCIR-LR)(l) over all of the l                    stored        -   If Ĉ_(NCIR-LR)(l)≧β            -   Storage of l and of Ĉ_(NCIR-LR)(l)            -   l=l+1            -   return to step 1

So as to give a more physical interpretation to the criterionĈ_(NCIR-LR)(l), we introduce the widely-linear space filterŴ(l)Δ[ŵ_(nc)(l)^(T), ŵ_(nc)(l)^(†)]^(T) defined by

Ŵ(l)Δ{circumflex over (R)}_(X)(l)⁻¹{circumflex over (r)}_(Xs)(l)  (13)

By considering that the sequence s(nT) is a particular form of a randomsignal, the expression (13) is none other than the estimate in the leastsquares sense of the widely-linear space filter, W(l)Δ[w_(nc)(l)^(T),w_(nc)(l)^(†)]^(T) ΔR_(X)(l)⁻¹r_(Xs)(l), which minimizes the mean squareerror (MSE) between the signal s(nT) and the real outputW^(†)X((l/p+n)T)=2Re[w^(†)x((l/p+n)T)], where WΔ[w^(T), w^(†)]^(T),R_(X)(l)ΔE[X((l/p+n)T)X((l/p+n)T)^(\)] andr_(Xs)(l)ΔE[X((l/p+n)T)s(nT)*]. In these conditions, the criterionĈ_(NCIR-LR)(l), defined by (10) where l_(o) has been replaced by l,takes the following form:

$\begin{matrix}{{{{\hat{C}}_{{NCIR}\text{-}{LR}}(l)}\underset{\_}{\Delta}\frac{\left( {1/K} \right){\sum\limits_{n = 0}^{K - 1}{{y_{nc}\left( {\left( {{l/p} + n} \right)T} \right)}{s({nT})}^{*}}}}{\left( {1/K} \right){\sum\limits_{n = 0}^{K - 1}{{s({nT})}}^{2}}}} = \frac{\sum\limits_{n = 0}^{K - 1}{{y_{nc}\left( {\left( {{l/p} + n} \right)T} \right)}{s({nT})}^{*}}}{\sum\limits_{n = 0}^{K - 1}{{s({nT})}}^{2}}} & (14)\end{matrix}$

where y_(nc)((l/p+n)T) corresponds to the output,y_(nc)((l/p+n)T)ΔŴ(l)^(†)X((l/p+n)T)=2Re[ŵ_(nc)(l)^(†)x((l/p+n)T)], ofthe filter Ŵ(l) the input of which is X((l/p+n)T).

In these conditions, the sufficient statistic Ĉ_(NCIR-LR)(l)corresponds, plus or minus the normalization factor, to the result ofthe correlation between the training sequence and the output of the LSLspace filter Ŵ(l), as is illustrated in FIG. 1.

Thus, as long as l is not close to l_(o), the sequence s(nT) is weaklycorrelated with the observation vector X((l/p+n)T), the vector W(l) isnot very far from the zero vector and the function Ĉ_(NCIR-LR)(l)approaches zero plus or minus the variance noise due to the finiteduration of the training sequence.

Conversely, at the synchronization instant l=l_(o), the sequence s(nT)is perfectly correlated with the useful part of the observation vectorX((l/p+n)T) given, under H1, by:

H1: X((l _(o) /p+n)T)≈s(nT)H _(s) +B _(T)((l _(o) /p+n)T)  (15)

and the vector r_(Xs)(l) becomes proportional to H_(s). Then, the vectorW(l) becomes proportional to the widely-linear space-matched filter,W_(s)(l)Δ[w_(nc,s)(l)^(T), w_(nc,s)(l)^(†)]^(T) ΔR_(X)(l)⁻¹H_(s), whichcorresponds to the widely-linear space filter which maximizes the signalto interference plus noise ratio (SINR) at the output. It is easy tocheck that this widely-linear space-matched filter also corresponds tothe conventional space-matched filter but for a virtual array of 2Nsensors receiving a useful signal for which the channel vector is H_(s)and an observed total vector noise B_(T)((l_(o)/p+n)T) at the time(l_(o)/p+n)T.

One consequence of this result is that the widely-linear space-matchedfilter W_(s)(l) is capable of rejecting up to P=2N−1 rectilinearinterferences from an array of N sensors and, more particularly, P=1rectilinear interference from a single sensor, hence the concept ofsingle antenna interference cancellation (SAIC). Then, when K increases,the criterion Ĉ_(NCIR-LR)(l) for l=l_(o) approaches the quantityC_(NCIR-LR)(l_(o)) given by:

$\begin{matrix}{{{C_{{NCIR} - {LR}}\left( l_{o} \right)}\underset{\_}{\Delta}\frac{{r_{Xs}\left( l_{o} \right)}^{\dagger}{R_{X}\left( l_{o} \right)}^{- 1}{r_{Xs}\left( l_{o} \right)}}{\pi_{s}}} = \frac{\left\lbrack {{SIN}\; R} \right\rbrack_{nc}\left( l_{o} \right)}{1 + {\left\lbrack {{SIN}\; R} \right\rbrack_{nc}\left( l_{o} \right)}}} & (16)\end{matrix}$

where [SINR]_(nc)(l_(o)) is the SINR at the output of the widely-linearspace-matched filter W_(s)(l_(o)) at the sampling instant l_(o)T_(e),defined by:

[SINR]_(nc)(l _(o))=π_(s) H _(s) R _(B)(l _(o))⁻¹ H _(s)  (17)

Assuming a total Gaussian and orthogonally sequenced noise, theprobability of correct synchronization is directly linked to the valueof the parameter ρ_(nc)=K [SINR]_(nc)(l_(o)), which is none other thanthe SINR at the correlation output, just before the comparison with thethreshold. This result remains valid in the presence of rectilinearinterferences.

Performance Characteristics

It is assumed that the total noise consists of a rectilinearinterference and a background noise. In these conditions, the noisevector b_(T)((l_(o)/p+n)T) takes the form:

b _(T)((l _(o) /p+n)T)≈j ₁((l _(o) /p+n)T)h ₁ +b((l _(o) /p+n)T)  (18)

where b((l_(o)/p+n)T) is the background noise vector at the instant((l_(o)/p+n)T), assumed to be centered, stationary and spatially white,h₁ is the channel vector of the interference and j₁((l_(o)/p+n)T) is thecomplex envelope of the interference at the instant ((l_(o)/p+n)T). Inthese conditions, the extended observation vector for the instant((l_(o)/p+n)T), X((l_(o)/p+n)T), can be expressed

X((l _(o) /p+n)T)≈s(nT)H _(s) +j ₁((l _(o) /p+n)T)H ₁ +B((l _(o)/p+n)T)  (19)

where B((l_(o)/p+n)T)Δ[b((l_(o)/p+n)T)^(T), b((l_(o)/p+n)T)^(†)]^(T), H₁Δ[h₁ ^(T), h₁ ^(†)]^(T) and where the matrices R(k) and C(k), areexpressed:

R(k)≈π₁(k)h ₁ h ₁ ^(\)+η₂ I  (20)

C(k)≈π₁(k)h ₁ h ₁ ^(T)  (21)

where η₂ is the average power of the background noise for each sensor, Iis the identity matrix (N×N) and π₁(k)ΔE[|j₁(kT_(e))|²] is the power ofthe interference received by an omnidirectional sensor for a propagationin free space.

Given the above assumptions, the spatial correlation coefficient,α_(1s,v), between the interference and the useful signal for the virtualarray of 2N sensors, defined by the normalized scalar product of thevectors H_(s) and H₁, and such that 0≦|α_(1s,v)|≦1, is given by:

$\begin{matrix}{{\alpha_{{1s},v}\underset{\_}{\Delta}\frac{H_{1}^{\dagger}H_{s}}{\left( {H_{1}^{\dagger}H_{1}} \right)^{1/2}\left( {H_{s}^{\dagger}H_{s}} \right)^{1/2}}} = {{\alpha_{1s}}\cos \; \psi}} & (22)\end{matrix}$

where ψ is the phase of h_(s) ^(\)h₁ and where α_(1s), such that0≦|α_(1s)|≦1, is the spatial correlation coefficient between theinterference and the useful signal for the real array of N sensors,defined by:

$\begin{matrix}{\alpha_{1s}\underset{\_}{\Delta}\frac{h_{1}^{\dagger}h_{s}}{\left( {h_{1}^{\dagger}h_{1}} \right)^{1/2}\left( {h_{s}^{\dagger}h_{s}} \right)^{1/2}}\underset{\_}{\Delta}{\alpha_{1s}}^{- {j\psi}}} & (23)\end{matrix}$

The expression (22) shows that the virtual array associated with anarray with space diversity is an array with space and phase diversity.Similarly, the virtual array associated with an array with space,pattern and polarization diversity is an array with space, pattern,polarization and phase diversity. One consequence of this result, provenby the fact that |α_(1s,v)|=|α_(1s)||cos ψ|≦|α_(1s)|, is that thewidely-linear space-matched filter (FAS LSL) discriminates the sourcesbetter than the space-matched filter and makes it possible in particularto reject single-sensor interference by phase discrimination. Inparticular, the SINR at the output of the widely-linear space-matchedfilter W_(s)(l_(o)), defined by (17), takes the form:

$\begin{matrix}{{\left\lbrack {{SIN}\; R} \right\rbrack_{nc}\left( l_{o} \right)} = {2{ɛ_{s}\left\lbrack {1 - {\frac{2ɛ_{1}}{1 + {2ɛ_{1}}}{\alpha_{1s}}^{2}\cos^{2}\psi}} \right\rbrack}}} & (24)\end{matrix}$

where ε_(s) Δ(h_(s) ^(\)h_(s))π_(s)/η₂ and ε₁ Δ(h₁ ^(†)h₁)π₁/η₂. Theexpression (24) shows that [SINR]_(nc)(l_(o)) is a decreasing functionof cos²ψ, |α_(1s)|² and ε₁, taking its minimum value in the absence ofspatial discrimination between the useful signal and the interference(|α_(1s)|=1), which is produced in particular for a single-sensorreception. In these conditions, for a strong interference (ε₁>>1), theexpression (24) takes the form:

[SINR]_(nc)(l _(o))≈2ε_(s)[1−cos²ψ]  (25)

an expression independent of ε₁, controlled by 2ε_(s) and cos²ψ, andshowing a capacity to reject rectilinear interference by phasediscrimination as long as ψ≠0+kπ, i.e. as long as there is a phasediscrimination between the useful signal and the interference, with adegradation of the performance characteristics compared to the situationin the absence of interference increasing with cos²ψ.

Examples of favorable, unfavorable and intermediate situations relatingto the differential phase ψ are illustrated in FIG. 2 for a propagationin free space and a BPSK interference.

Furthermore, FIG. 3 illustrates the operation of the widely-linearspace-matched filter in the presence of a strong BPSK interference,which compensates the phase of the interference and phase shifts that ofπ/2 so as to minimize the contribution of the interference on the realpart axis.

According to one variant of embodiment, the method according to theinvention is applied in particular to the GMSK modulation belonging tothe family of continuous phase modulations (CPM). It is shown in [6]that the GMSK modulation can be approximated by a linear modulation,generating the approximate useful complex envelope:

$\begin{matrix}{{s(t)} = {\mu_{s}{\sum\limits_{n}{j^{n}b_{n}{f\left( {t - {nT}} \right)}}}}} & (26)\end{matrix}$

where b_(n)=±1 are random i.i.d variables corresponding to thetransmitted symbols if the symbols are differentially encoded in theexact form of the modulation, T is the symbol duration and f(t) the realvalue shaping filter which corresponds either to the main pulse in theLaurent breakdown or the best pulse in the least squares sense, forexample. In both cases, the time bearer of f(t) is approximately 4T andthe sampled version of f(t) at the symbol rate generates only threenon-zero values corresponding to f(0), the maximum value of f(t), andtwo non-zero secondary values, f(T) and f(−T), such thatf(T)=f(−t)<f(0). The derotation operation involves multiplying thesample, s(nT), of s(t) by j^(−n), generating the derotated sampledsignal, s_(d)(nT), defined by:

$\begin{matrix}{{{s_{d}({nT})}\underset{\_}{\Delta}j^{- n}{s({nT})}} = {\mu_{s}{\sum\limits_{m}{j^{m - n}b_{m}{f\left( {\left( {n - m} \right)T} \right)}\underset{\_}{\Delta}\mu_{s}{\sum\limits_{m}{b_{m}{f_{d}\left( {\left( {n - m} \right)T} \right)}}}}}}} & (27)\end{matrix}$

where f_(d)(t)Δj^(−t/T)f(t) is the equivalent shaping filter of thelinearized and derotated GMSK signal. It can be deduced from (27) thats_(d)(nT) has the form of a BPSK signal sampled at the symbol rate butwith two differences compared to BPSK. The first lies in the fact thatf_(d)(t) is not a ½ Nyquist filter and that the inter-symbolinterference (ISI) appears after a filtering operation matched to thefilter f_(d)(t). The second lies in the fact that f_(d)(t) is no longera function with real values but becomes a function with complex values.

Extended Observation Vector

To simplify the analysis, a useful signal and a GMSK interference thatare synchronized are considered. In these conditions, the observationvector sampled at the symbol rate and derotated is expressed, for thesynchronization instant l_(o)T_(e),

x _(d)((l _(o) /p+n)T)Δ j ^(−n) x((l _(o) /p+n)T)≈j ^(−n) s(nT)h _(s) +j^(−n) j ₁(nT)h ₁ +j ^(−n) b((l _(o) p+n)T)  (28)

By inserting (27) into (28), we obtain:

x _(d)((l _(o) /p+n)T)≈μ_(s) [f(0)b _(n) +jf(−T)b _(n+1) −jr(T)b _(n−1)]h _(s)+μ₁ [f(0)b _(n) ¹ +jf(−T)b _(n+1) ¹ −jr(T)b _(n−1) ¹ ]h ₁ +j^(−n) b((l _(o) /p+n)T)  (29)

where μ₁ controls the amplitude of the interference and where b_(n) ¹ isthe symbol n of the interference. From (29), it is possible to deducethe expression of the extended derotated observationX_(d)((l_(o)/p+n)T)Δ[x_(d)((l_(o)/p+n)T)^(T),x_(d)((l_(o)/p+n)T)^(†)]^(T), given by:

X _(d)((l _(o) /p+n)T)≈μ_(s) f(0)b _(n) H _(s)+μ_(s) [f(−T)b _(n+1)−f(T)b _(n−1) ]JH _(s)+μ₁ f(0)b _(n);¹ H1+μ₁[(−T)b _(n);¹ ₊₁ −f(T)b_(n);¹ ⁻¹ ]JH ₁ +B _(d)((l _(o) /p+n)T)  (30)

where B_(d)((l_(o)/p+n)T)Δ[j^(−n)b((l_(o)/p+n)T)^(T),j^(n)b((l_(o)/p+n)T)^(†)]^(T) and J is the matrix (2N×2N) defined by:

$\begin{matrix}{J\; \underset{\_}{\Delta}{j\begin{pmatrix}I & O \\O & {- I}\end{pmatrix}}} & (31)\end{matrix}$

where I and O are respectively the identity and zero matrices (N×N).Comparing (30) and (19), it can be deduced that, unlike the BPSKsources, a derotated GMSK source i (useful (i=s) or interfering (i=1))generates, in the extended observation vector, X_(d)((l_(o)/p+n)T), twostatistically independent sources of powers π_(i1) Δμ_(i) ²f(0)² andπ_(i2) Δμ_(i) ²[f(−T)²+f(T)²] and of channel vectors given respectivelyby H_(i) and JH_(i), such that H_(i) ^(†)JH_(i)=0.

Limitations of the Widely-Linear Space Filters

It can be deduced from the above result that two degrees of freedom arenecessary to process a derotated GMSK interference fromX_(d)((l_(o)/p+n)T). Thus, while considering space filters(y((l_(o)/p+n)T)ΔW^(†)X_(d)((l_(o)/p+n)T)), the number of virtualsensors must remain at least greater than the number of interferencesgenerated (2N>2P), which eliminates the interest of the optimalwidely-linear filters. However, in as much as the two interferencesgenerated in (30) are two different filtered versions of one and thesame source, the problem of the rejection of these two interferences issimilar to the problem of the rejection of an interference having passedthrough a multiple-path propagation channel. In these conditions, theproblem can be resolved by replacing the widely-linear space filterswith widely-linear space-time filters.

Synchronization Using Widely-Linear Space-Time Filters

A widely-linear space-time filter (ST LSL) with L coefficients for eachfilter generates, at the synchronization instant l_(o)T_(e), the outputy((l_(o)/p+n)T) defined by:

$\begin{matrix}{{{y\left( {\left( {{l_{o}/p} + n} \right)T} \right)}\underset{\_}{\Delta}{\sum\limits_{q = {{- {({L - 1})}}/2}}^{{({L - 1})}/2}{W_{q}^{\dagger}{X_{d}\left( {\left( {{l_{o}/p} + n - q} \right)T} \right)}\underset{\_}{\Delta}W_{st}^{\dagger}{X_{d,{st}}\left( {\left( {{l_{o}/p} + n} \right)T} \right)}}}}{{if}\mspace{14mu} L\mspace{14mu} {is}\mspace{14mu} {odd}\mspace{14mu} {and}}} & (32) \\{{y\left( {\left( {{l_{o}/p} + n} \right)T} \right)}\underset{\_}{\Delta}{\sum\limits_{q = {{- L}/2}}^{{L/2} - 1}{W_{q}^{\dagger}{X_{d}\left( {\left( {{l_{o}/p} + n - q} \right)T} \right)}\underset{\_}{\Delta}W_{st}^{\dagger}{X_{d,{st}}\left( {\left( {{l_{o}/p} + n} \right)T} \right)}}}} & (33)\end{matrix}$

if L is even, where the vectors (2LN×1)W_(st) and X_(d,st)((l_(o)/p+n)T)are defined respectively by W_(st) Δ[W_(−(L−1)/2) ^(T), . . . ,W_((L−1)/2) ^(T)]^(T) andX_(d,st)((l_(o)/p+n)T)Δ[X_(d)((l_(o)/p+n+(L−1)/2)T)^(T), . . . ,X_(d)((l_(o)/p+n−(L−1)/2)T)^(T)]^(T) if L is odd and W_(st) Δ[W_(−L/2)^(T), . . . , W_(L/2−1) ^(T)]^(T) andX_(d,st)((l_(o)/p+n)T)Δ[X_(d)((l_(o)/p+n+L/2)T)^(T), . . . ,X_(d)((l_(o)/p+n−L/2+1)T)^(T)]^(T) if L is even. In these conditions,the proposed procedure for synchronizing a GMSK signal in the presenceof GMSK interferences is similar to that proposed for the BPSK signalsin section 5, but where the widely-linear space filter (2N×1)Ŵ(l),defined by (13), is replaced by the widely-linear space-time filter(2LN×1), Ŵ_(st)(l), defined by:

Ŵ_(st)(l)Δ{circumflex over (R)}_(Xd,st)(l)⁻¹{circumflex over(r)}_(Xd,st,s)(l)  (34)

where the matrix (2LN×2LN){circumflex over (R)}_(Xd,st)(l) and thevector (2LN×1){circumflex over (r)}_(Xd,st,s)(l) are definedrespectively by:

$\begin{matrix}{{{\hat{R}}_{{Xd},{st}}(l)}\underset{\_}{\Delta}\frac{1}{K}{\sum\limits_{n = 0}^{K - 1}{{X_{d,{st}}\left( {\left( {{l/p} + n} \right)T} \right)}{X_{d,{st}}\left( {\left( {{l/p} + n} \right)T} \right)}^{\dagger}}}} & (35) \\{{{\hat{r}}_{{Xd},{st},s}(l)}\underset{\_}{\Delta}\frac{1}{K}{\sum\limits_{n = 0}^{K - 1}{{X_{d,{st}}\left( {\left( {{l/p} + n} \right)T} \right)}{s({nT})}^{*}}}} & (36)\end{matrix}$

In these conditions, the sufficient statistic tested at the instantlT_(e) is expressed:

$\begin{matrix}{{{{\,^{\hat{C}}{NCIR}} - {{{LR}(l)}{\underset{\_}{\Delta}\left( {1/K} \right)}{\sum\limits_{n = 0}^{K - 1}{{y_{{nc},{st}}\left( {\left( {{l/p} + n} \right)T} \right)}{s({nT})}^{*}}}}};} = {{\left( {1/K} \right){\sum\limits_{n = 0}^{K - 1}{{s({nT})}}^{2}}} = {\sum\limits_{n = 0}^{K - 1}{{y_{{nc},{st}}\left( {\left( {{l/p} + n} \right)T} \right)}{s({nT})}^{*}{\sum\limits_{n = 0}^{K - 1}{{s({nT})}}^{2}}}}}} & (37)\end{matrix}$

where y_(nc,st)((l/p+n)T) corresponds to the output,y_(nc,st)((l/p+n)T)ΔŴ_(st)(l)^(†)X_(d,st)((l/p+n)T), of the filterŴ_(st)(l), the input of which is X_(d,st)((l/p+n)T). This sufficientstatistic Ĉ_(NCIR-LR)(l) corresponds, plus or minus the normalizationfactor, to the result of the correlation between the training sequenceand the output of the widely-linear space-time filter Ŵ_(st)(l).

The expression (37) can also be expressed:

$\begin{matrix}{{{\hat{C}}_{{NCIR}\text{-}{LR}}(l)}\underset{\_}{\Delta}\frac{{{\hat{r}}_{{Xd},{st},s}(l)}^{\dagger}{{\hat{R}}_{{Xd},{st}}(l)}^{- 1}{{\hat{r}}_{{Xd},{st},s}(l)}}{\left( {1/K} \right){\sum\limits_{n = 0}^{K - 1}{{s({nT})}}^{2}}}} & (38)\end{matrix}$

The steps of the method for the GMSK signals are summarized below:

-   -   Step 0: Initialization l=l_(min) (l_(min)=0 for example) and        choice of the detection threshold β    -   Step 1: Derotation of the observations and construction of the        space-time observation vectors (2LN×1)X_(d,st)((l/p+n)T),        0≦n≦K−1    -   Step 2: Estimation of {circumflex over (r)}_(Xd,st,s)(l) and of        {circumflex over (R)}_(Xd,st)(l)    -   Step 3: Calculation of the sufficient statistic Ĉ_(NCIR-LR)(l),        defined by (38)    -   Step 4: Comparison of Ĉ_(NCIR-LR)(l) with the threshold β    -   Step 5: Decision        -   If Ĉ_(NCIR-LR)(l)<β            -   If Ĉ_(NCIR-LR)(l−1)<β                -   l=l+1                -   return to step 1            -   If Ĉ_(NCIR-LR)(l−1)≧β                -   The synchronization instant is ĺ_(o)T_(e) where                    l=ĺ_(o) maximizes Ĉ_(NCIR-LR)(l) over all of the l                    stored        -   If Ĉ_(NCIR-LR)(l)≧β            -   Storage of l and of Ĉ_(NCIR-LR)(l)            -   l=l+1            -   return to step 1

It is possible to show that, in the presence of P GMSK interferences, asufficient condition for rejecting all the interferences (including theinter-symbol interferences) at the output of the filter Ŵ_(st)(l_(o)) isthat the following condition should be met:

$\begin{matrix}{P < \frac{{L\left( {{2N} - 1} \right)} - 2}{L + 3}} & (39)\end{matrix}$

which generates the sufficient condition P<2N−1 for L that is infinitelygreat and which shows at the same time the possibility of processing,using the optimal widely-linear space-time filters (ST LSL) and for N>1,a number of GMSK interferences at least equal to 2(N−1) from N sensorsand therefore two times greater than could be envisaged with aconventional processing. However, this condition is only sufficient anddoes not take account, for example, of the fact that the correlationoperation between the output y_(nc,st)((l_(o)/p+n)T) and the sequences(nT) provides an additional SINR gain of the order of K. Because ofthis, a rejection under the background noise of all the interferencesources at the output of the filter Ŵ_(st)(l_(o)) is not necessarilyneeded on synchronization. In these conditions, a limited number ofcoefficients (L=3 or 5) can be used to obtain very good results innumerous situations inherent to the GSM network context, including forN=P=1.

The optimal widely-linear space-time filters can also prove advantageousfor synchronizing BPSK signals, particularly when the propagation delayis not a multiple of the sampling period.

The method described above applies for all types of propagation channels(time-spread or not).

REFERENCES

-   [1] S. BENSLEY, B. AAZHANG, “Subspace-based channel estimation for    CDMA system, IEEE Trans Communication, Vol 44, pp. 1009-1020, August    1996-   [2] S. BENSLEY, B. AAZHANG, “Maximum Likelihood synchronization of a    single user for CDMA systems, IEEE Trans Communication, Vol 46, pp.    392-399, March 1998-   [3] L. E. BRENNAN, I. S. REED, “An adaptive array signal processing    algorithm for communications”, IEEE Trans. Aerosp. Electronic    Systems, Vol 18, No 1, pp. 124-130, January 1982.-   [4] R. T. COMPTON, “An adaptive array in a spread spectrum    communication system”, Proc IEEE, Vol 66, No 3, pp. 289-298, March    1978-   [5] D. M. DUGLOS, R. A. SCHOLTZ, “Acquisition of spread spectrum    signals by an adaptive array”, IEEE Trans. Acou. Speech. Signal    Proc., Vol 37, No 8, pp. 1253-1270, August 1989.-   [6] P. A. LAURENT, “Exact and approximate construction of digital    phase modulations by superposition of amplitude modulated pulses    (AMP)”, IEEE Trans. on Communications, Vol 34, No 2, pp. 150-160,    February 1986.-   [7] B. PICINBONO, P. CHEVALIER, “Widely linear estimation with    complex data”, IEEE Trans. Signal Processing, Vol 43, No 8, pp.    2030-2033, August 1995.-   [8] H. L. VAN TREES, “Detection, Estimation and Modulation Theory”,    John Wiley and Sons, 1971.-   [9] J. H. WINTERS, “Spread spectrum in a four phase communication    system employing adaptive antennas”, IEEE Trans. On Communications,    Vol 30, No 5, pp. 929-936, May 1982.

1. A method of synchronizing a substantially rectilinear signal beingpropagated through an unknown channel, in the presence of unknownsubstantially rectilinear interferences, received by an array of Nsensors, in which a known training sequence s(nT) is used comprising Ksymbols and sampled at a symbol rate T(s(nT), 0≦n≦{tilde over (K)}1)wherein, based on observations x((n+l/p)T) over the duration of thetraining sequence, where p=T/Te is an integer and Te the samplingperiod, a virtual observation vector X((n+l/p)T)=[x((n+l/p)T)T,x((n+l/p)T)†]T is defined, as well as a decision criterion or decisionstatistic taking into account the second-order non-circular nature ofthe interferences, by using the first and second correlation matrices ofthe virtual observation vector X((n+l/p)T).
 2. The synchronizationmethod as claimed in claim 1, comprising the following steps, forrectilinear signals: acquiring observation vectors (N×1), x((l/p+n)T),0≦n≦{tilde over (K)}1, where T=pT_(e) is the symbol duration, p is aninteger and Te the sampling period, l being the delay of the signal,constructing virtual observation vectors (2N×1),X((n+l/p)T)=[x((n+l/p)T)T, x((n+l/p)T)†]T, choosing L as the number oftime coefficients and constructing virtual space-time observationvectors (2LN×1), X_(st)((n+l/p)T)Δ[X((l/p+n+(L−1)/2)T)^(T), . . . ,X((l/p+ñ(L−1)/2)T)^(T)]^(T) if L is odd andX_(st)((l/p+n)T)Δ[X((l/p+n+L/2)T)^(T), . . . , X((l/p+ñL/2+1)T)^(T)]^(T)if L is even, determining the intercorrelation vector {circumflex over(r)}_(X,st,s(l)) and the correlation matrix {circumflex over(R)}_(X,st(l)) of the space-time observations, for the sampling instantl, taking into account the second correlation matrix of the vectorx((l/p+n)T), defining a space-time filter ST and its estimate from{circumflex over (R)}_(X,st(l)) and {circumflex over (r)}_(X,st,s(l)),such that Ŵ_(st(l)) Δ{circumflex over(R)}_(X,st({tilde over (l)})1){circumflex over (r)}_(X,st,s(l)),defining a synchronization criterion Ĉ_(NCIR-LR(l)) by correlating theoutput of the space-time filter Ŵ_(st(l)) and the training sequence,comparing the criterion Ĉ_(NCIR-LR)(l) with a threshold β set for agiven false alarm probability.
 3. The method as claimed in claim 1,comprising, for two-state CPM signals, a preprocessing step forderotating the received observations.
 4. The method as claimed in claim2, wherein:${{\hat{C}}_{{NCIR}\text{-}{LR}}(l)}\underset{\_}{\Delta}\frac{{{\hat{r}}_{X,{st},s}(l)}^{\dagger}{{\hat{R}}_{X,{st}}(l)}^{- 1}{{\hat{r}}_{X,{st},s}(l)}}{\left( {1/K} \right){\sum\limits_{n = 0}^{K - 1}{{s({nT})}}^{2}}}$where the matrix (2LN×2LN){circumflex over (R)}_(X,st(l)) and the vector(2LN×1){circumflex over (r)}_(X,st,s(l)) are defined respectively by:${{\hat{R}}_{X,{st}}(l)}\underset{\_}{\Delta}\frac{1}{K}{\sum\limits_{n = 0}^{K - 1}{{X_{st}\left( {\left( {{l/p} + n} \right)T} \right)}{X_{st}\left( {\left( {{l/p} + n} \right)T} \right)}^{\dagger}}}$${{{\hat{r}}_{X,{st},s}(l)}\underset{\_}{\Delta}\frac{1}{K}{\sum\limits_{n = 0}^{K - 1}{{X_{st}\left( {\left( {{l/p} + n} \right)T} \right)}{s({nT})}^{*}}}}\;$where 0≦Ĉ_(NCIR-LR(l))≦1.
 5. A device for synchronizing a substantiallyrectilinear signal being propagated through an unknown channel, in thepresence of unknown substantially rectilinear interferences, in an arrayof N sensors, in which a known training sequence s(nT) is usedcomprising K symbols and sampled at the symbol rate (s(nT), 0≦n≦K−1),comprising: a device able to determine a decision criterion or decisionstatistic from observations x((n+l/p)T) over the duration of thetraining sequence, where p=T/Te is an integer and Te the samplingperiod, a virtual observation vector X((n+l/p)T)=[x((n+l/p)T)T,x((n+l/p)T)†]T, and a decision criterion or decision statistic, takinginto account the second-order non-circular nature of the interferences,by using the first and second correlation matrices of the virtualobservation vector X((n+l/p)T).
 6. The device as claimed in claim 5,comprising an array of virtual sensors (N+1 to 2N), a filter (1), adevice for correlating the signal obtained from the filter and from thetraining sequence, and a decision device receiving the correlatedsignal.
 7. The device as claimed in claim 5, comprising a device able toconvert a two-state CPM signal into a quasi-rectilinear signal.
 8. Thedevice as claimed in claim 5 wherein the device synchronizes aone-dimensional modulation signal: ASK, BPSK, etc.
 9. The method asclaimed in claim 3, comprising synchronizing a two-state CPM signal ofMSK, GMSK or other type.
 10. The device as claimed in claim 7, whereinthe device synchronizes a two-state CPM signal of MSK, GMSK or othertype.
 11. The method as claimed in claim 1, comprising synchronizing aone-dimensional modulation signal: ASK, BPSK, etc.